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Chapter 3: Problem 30

Given the following functions, find the function values. \(f(x)=25 x-15,\) find \(f(3)\)

Short Answer

Expert verified
The value of \( f(3) \) is 60.

Step by step solution

01

Understand the Function

The given function is \( f(x) = 25x - 15 \). This is a linear function, where \( 25 \) is the coefficient of \( x \), and \( -15 \) is the constant term.
02

Substitute the given value into the function

We need to find \( f(3) \). To do this, substitute \( x = 3 \) into the function \( f(x) = 25x - 15 \). This means replacing every occurrence of \( x \) with \( 3 \).
03

Calculate the expression

Substitute \( x = 3 \) into the function: \( f(3) = 25(3) - 15 \). Now calculate the expression: \( 25 \times 3 = 75 \), and then subtract \( 15 \) to get \( 75 - 15 \).
04

Simplify the result

Simplify the result: \( 75 - 15 = 60 \). Thus, \( f(3) = 60 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Values
When we're talking about function values, we're essentially looking for the output of a function when a specific input is given. In our case, we have a function defined as \( f(x) = 25x - 15 \). Here, \( x \) is our input variable. When you input a particular number for \( x \), the function gives you a number back, which is called the function value.
  • For example, if we want to find \( f(3) \), it means putting 3 in place of \( x \) in our function.
  • This replaces every \( x \) in \( f(x) \) with 3.
  • Function values help us understand what the function outputs at specific points.
Grasping how to find function values through substitution is a core skill in math problems involving functions.
Substitution Method
The substitution method is like a simple plug-and-play process in mathematics. This technique helps you find solutions by replacing variables with given numbers. Let's break it down using our function \( f(x) = 25x - 15 \).
  • Imagine you have the task \( f(3) \): all you need to do is substitute \( x = 3 \) into the function.
  • Whenever you see \( x \) in the expression, replace it with 3, transforming the function into \( f(3) = 25(3) - 15 \).
  • This way, you're using specific values to get the exact output or function value.
Using substitution not only makes calculations more straightforward but also bridges the gap between the abstract formula and concrete numbers you can work with. It turns continuous functions into understandable, discrete outputs.
Simplifying Expressions
Once you've carried out substitution, your next move is simplifying. Simplifying expressions means reducing them to their simplest form. This makes them easier to understand and work with.In our example, after substituting 3 into the function \( f(x) = 25x - 15 \), we get \( f(3) = 25(3) - 15 \).
  • First, compute \( 25 \times 3 \), which equals 75.
  • Next, subtract 15 from 75, which simplifies to 60.
  • So, \( f(3) = 60 \).
Simplifying helps tidy up the expression, giving you a clear, concise answer without extra steps to worry about. It's like clearing the path to see the solution more clearly鈥攊t involves arithmetic operations and logical steps to bring everything together neatly.

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